In all cases, if we start from initial conditions such that the entropy nearly vanishes at \(t=0\) , we observe that, for all values of \(q\) , \(S_q\)  tends to increase (not necessarily in a monotonic manner) as time increases. But it tends to increase linearly (thus providing a finite entropy production per unit time) only for an unique value of the index \(q\) .
For \(a=2\) , the entropy which linearly increases with time, thus yielding a finite entropy production per unit time (satisfying the Pesin identity for the entropy production per unit time \(K_{BG} \equiv \lim _{t\rightarrow \infty } S_{BG}(t)/t =\lambda \) ), is \(S_{BG}\) . In contrast, at the edge of chaos, the entropy which linearly increases with time is \(S_q\)  with \(q=q_c \equiv 0.24448770134128\dots \)  (see [1]}, [2]} and references therein).
